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2

The Heston model can have that property. If you make the correlation negative between the Brownian motions in the $dS_{t}$ process and the $d\nu_{t}$ process you imply that price is negatively correlated with variance.

1

Yeah this is often called Spot-Vol correlation and is well known. Most people take this into account. I think if you just google spot-vol correlation you will come up with many example/models.

1

Apologies for the delay on the hedging of non-forward-starting volatility swaps, but it's only since this week that I have an answer for this. So, for plain volswaps, I can give you a nonparametric hedge in terms of varswaps only. That's not as cheap as using a single option (which I believe is not possibe anyway), but certainly better than trading an ...

7

I've seen that Gordon answer is more concise and to the point. Take this as a complementary answer. This is a general approach that will work for all this type of linear SDEs, not just this one. Assume we have the following linear SDE $$dX_t = (F_t X_t +f_t)dt + (G_t X_t +g_t)dB_t \tag*{(1)}$$ where $F, G, f$ and $g$ are Borel measurable bounded ...

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Let \begin{align*} Y_t = e^{(a+\frac{c^2}{2})t-cW_t}. \end{align*} Then \begin{align*} dY_t = Y_t\left[\big(a+c^2\big)dt -c dW_t \right]. \end{align*} Moreover, \begin{align*} d(X_tY_t) &= Y_t dX_t + X_t dY_t + d\langle X, Y\rangle_t\\ &=abY_tdt. \end{align*} That is, \begin{align*} X_t = Y_t^{-1}\left(X_0 + ab\int_0^t Y_sds\right). \end{align*}

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